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This is a corrigendum to our paper, “On a class of prime entire functions”, published in Acta Math. Sin., Engl. Ser., 25, 1647–1652 (2009).

In this work, some counterexamples are given to refute some results reported in the paper by Guo and Li [8] (J Optim Theory Appl 162,(2014), 821-844). We correct the faulty in some of their theorems and we present alternative proofs. Moreover, we extend the definition of approximately pseudo-dissipative in the setting of metrizable topological vector spaces.

In this note, we give affirmative answers to two open problems posed by Agarwal et al. in the paper (J Optim Theory Appl 163(2):482---490, 2014).

With this note, we want to open a discussion on the duality theory for Interval Optimization. We begin by showing that some duality results Wu ( J Optim Theory Appl 144:615---628, 2010) are useless.

McCormick (Math Prog 10(1):147–175, 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F ◦ f , where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F , and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormicks result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed. Congratulations to the authors of the awardedpapers! Iwould also like to thankSpringer for sponsoring this award and the committee members for their active involvement. Nominations for the 2015 JOGO Best Paper Award should be emailed to butenko@tamu.edu. All JOGO papers published in 2015 (volumes 61–63) are eligible.

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

*Correspondence: wadade@se.kanazawa-u.ac.jp 3Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan Full list of author information is available at the end of the article The purpose of this note is to clarify the novelty of the paper entitled ‘Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces’ which was published in the J. Inequal. Appl. :,  []. After this paper was published, the authors were informed of the references [–], and [], the results of which partly overlap with those of []. In the paper [], the authors established the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces H n p p,q(R); see Section  in [] for the precise definition of H n p p,q(R). The main theorem in [] is stated as follows. Theorem A [, Theorem .] Let n ∈ N,  < p

Each year, the Computational Optimization and Applications (COAP) editorial board selects a paper from the preceding year’s COAP publications for the Best Paper Award. The recipient of the award for papers published in 2007 is Olvi Mangasarian of the University of Wisconsin, Madison and the University of California, San Diego, for his paper “Absolute Value Programming”, published in Volume 36, pages 43–53. This paper [7] as well as subsequent closely related papers [6, 8, 9] deal with the absolute value equation (AVE) Ax + B|x| = b, where A and B are arbitrary m× n real matrices and |x| denotes the vector with absolute values of the n-dimensional real valued vector x. The significance of this class of NP-hard problems arises partly from the fact that when B = I , the identity matrix, AVE is equivalent to the general linear complementarity problem, 0 ≤ x ⊥Mx + q ≥ 0. Even though problems involving absolute values are NP-hard, they share some very interesting properties with those of linear systems. For example, optimization problems with absolute value constraints possess optimality and duality results similar to those of linear programming, even though the problems are inherently nonconvex. Another interesting property that AVE shares with linear inequalities are theorems of the alternative which are established in this paper. The paper also contains a finite successive linearization algorithm for solving absolute value equations that terminates at a necessary optimality condition. This algorithm has solved all random test problems given to it for which mostly m ≥ 2n or n ≥ 2m, up to size (m,n) of (2000,100) and (100,2000). When m = n and B is invertible, which is the case for the linear complementarity problem formulation, a simpler concave minimization

In this note we correct an error in the paper (Ariza-Ruiz et al. in J Optim Theory Appl 167:409–429, 2015).

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles.

Zannier proved that for a Laurent series f(x) satisfying the functional equation of type f(x m ) = P(x, f(x)), where \( P(x, y) \in {\Bbb C}(x)[y] \), if f(x) is not rational the set of such m consists of the powers of a single integer. He mentioned that the case f(x) = P(x, f(x m )) should be proved in a similar way. In this paper we first verify this statement and second we show a theorem which is useful for proving the transcendence of a Laurent series satisfying a certain functional equation. This theorem is a generalization of the result that a Laurent series which satisfiesf(x m ) = P(x, f(x)), where\( P(x, y) \in {\Bbb C}(x, y),\,m \geq 2 \)cannot represent an algebraic function unless it is rational (Ke. Nishioka [1], Zannier [8], Section 3).

In Sec. 1 a correction is given of the estimate of the Hausdorff dimension and an estimate of the fractal dimension of a bounded subset of a Hilbert space, semiinvariant with respect to a flattening transformation. In Sec. 2 the results, proved by the author for semigroups with a continuous group parameter t∈R+≡[0, ∞), are carried over to the case when t runs through the semigroup ℑ+≡{t∈ℑ∣t⩾0} of some additive group ℑ⊂R=(−∞, ∞).

Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let t∈L1(0,1), let Nt be the smallest rearrangement-invariant order ideal vector lattice in L1(0,1), containing t. In the paper one investigates the properties of the image E(Nt¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, g∈L1(0,1), such that E(Nt¦A)⊂Ng. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(t∣A)≺QE(t*∣A*), where ≺ is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1⩽Q⩽25. One formulates the problem of the smallest value of Q.

In this paper, we study the dynamics of an infectious disease in the presence of a continuous-imperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E$$\end{document}RE. Then, we obtain the threshold dynamics of the system with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E$$\end{document}RE by studying the global stability of the disease-free equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E1$$\end{document}RE>1. For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_E

It is shown that one of the results of Schaefer can be obtained in a more elementary way and can be improved. This is an immediate corollary of a paper of the author Duke Math. J.19, 75---91 (1952).

In this short paper, we determine the minimal generating set of 1-generator generalized quasi-cyclic codes over ź4$\mathbb {Z}_{4}$. We also determine their rank and introduce a lower bound for the minimum distance of free 1-generator generalized quasi-cyclic codes. Further, we construct some new ź4$\mathbb {Z}_{4}$-linear codes and we obtain some good binary nonlinear codes using the usual Gray map.

In this short paper, we present two techniques to perform residue number system (RNS) to binary conversion using diagonal function and show the relationship between the techniques for RNS to binary conversion using Chinese remainder theorem and diagonal function. We also consider RNS to binary conversion using another monotonic function due to Pirlo and Impedovo.

Let $$p$$ p and $$q$$ q be two odd primes with $$p=Mf+1$$ p = Mf + 1 and $$M$$ M is even. A new construction of $$M$$ M -ary sequences of period $$pq$$ pq with low periodic autocorrelation is presented in this paper based on interleaving the $$M$$ M -ary power residue sequence of period $$p$$ p according to the quadratic residue with respect to $$q$$ q . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if $$M=2$$ M = 2 . For $$M=4$$ M = 4 , a new class of quaternary sequences of period $$pq$$ pq with maximal nontrivial autocorrelation value being either $$\sqrt{5}$$ 5 or $$3$$ 3 is obtained. This achieves the best known results for such kind of quaternary sequences.

Linear codes with good parameters have wide applications in secret sharing schemes, authentication codes, association schemes, consumer electronics and communications, etc. During the past four decades, constructions of linear codes with good parameters received much attention and many classes of such codes were presented. In this paper, we obtain a family of linear codes with good parameters over $$\mathbb {F}_p$$ by exploring further properties of constant dimension subspace codes, where p is a prime. The weight distribution of three classes of linear codes presented in this family is determined. Most notably, three classes of linear codes presented in this family are distance-optimal with respect to the Griesmer bound. Also, this paper presents a sufficient and necessary condition for this family of linear codes to have a $$\lambda $$ -dimensional hull. In addition, we show that our linear codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs.

Solomon and Elkin (SIAM J Discret Math 28(3):1173–1198, 2014) constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight $$(1+\epsilon )$$ -spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after $$O(\log n)$$ hops and requires $$O(\varDelta \log n)$$ bits of storage per vertex where $$\varDelta $$ is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.

We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form $\{(x,F(x)) \ : \ x\in {\mathbb {F}}_{2^{n}}\}$ { ( x , F ( x ) ) : x ∈ F 2 n } where F is a vectorial Boolean function mapping ${\mathbb {F}}_{2^{n}}$ F 2 n to ${\mathbb {F}}_{2^{n}}$ F 2 n .

The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

In this paper, we investigate the existence of value for a two-person zero-sum differential game with symmetric incomplete information and with signal revealing. Before the game begins, the initial state of the dynamic is chosen randomly among a finite number of points in $$\mathbb {R}^n$$ , while both players have only a probabilistic knowledge of the chosen initial state. During the game, if the system reaches a fixed closed target set K, the current state of the system at the hitting time is revealed to both players. We prove in this paper that this game has a value and its value function is the unique bounded continuous viscosity solution of a suitable Hamilton–Jacobi–Isaacs equation.

This paper is concerned with the identification of important nodes in node-weighted graphs by applying matrix functions, in particular the matrix exponential. Many tools that use an adjacency matrix for a graph have been developed to study the importance of the nodes in unweighted or edge-weighted networks. However, adjacency matrices for node-weighted graphs have not received much attention. The present paper proposes using a line graph associated with a node-weighted graph to construct an edge-weighted graph that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node increases, the importance of this node in the graph increases as well, provided that the adjacency matrix is suitably scaled. Applications to real-life problems are presented.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, two families of binary subfield codes with a few weights are presented from two special classes of linear codes, and their parameters are explicitly determined. Moreover, the parameters of the duals of these subfield codes are also studied. The two infinite families of subfield codes presented in this paper are distance-optimal with respect to the Griesmer bound and their duals are almost distance-optimal with respect to the sphere-packing bound.

A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $L^2(\mathbb{R})$, which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $\mathbb{Z}$, which are orbits of bounded operators on $L^2(\mathbb{R})$. Two classes of overcomplete Gabor frames which cannot be ordered over $\mathbb{Z}$ and represented by orbits of operators in $GL(L^2(\mathbb{R}))$ are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames., Comment: 12 pages

Many cryptographic protocols which are based on elliptic curves require to efficiently encode bit-strings into the points of a given elliptic curve such that the encoding function satisfies computability, regularity, and samplability, or generally admissibility. All the admissible encoding functions from the finite field $$\mathbb {F}_q$$ are restricted to the class of elliptic curves with a non-trivial l-torsion $$\mathbb {F}_q$$ -rational point, where $$l\in \{2,3\}$$ . Therefore, there is no admissible encoding function to the class of many cryptographically interesting elliptic curves of prime order. In this paper, we present an admissible 2:1 encoding function from the set $$\{0,1,\ldots , \frac{q-1}{2}\}$$ to the $$\mathbb {F}_q$$ -rational points of arbitrary elliptic curves. We also propose an injective encoding function to elliptic curves with a non-trivial $$\mathbb {F}_q$$ -rational point of order two, that acts the same as the Bernstein et al.’s injective encoding function. Conversely, occasionally we have to transmit points of a known curve through an insecure channel. Traditional methods for transferring points enable an adversary to recognize patterns in the transmitted data. Consequently, one finds valuable information to attack the cryptosystem using the network traffic. By the help of the inverse of the injective encoding functions, Bernstein et al. introduced an interesting solution to this problem, namely Elligator. In this paper, we present an indistinguishable elliptic curve point representation using our given encoding function, which unlike the previous well-known encoding functions is not injective but covers almost all of elliptic curves over odd characteristic finite fields. Indeed, since we proposed a 2:1 encoding function to elliptic curves in short Weierstrass form, we have to select one pre-image randomly and transmit its corresponding bit-string instead of the point.

RS-like locally recoverable (LRC) codes have construction based on the classical construction of Reed–Solomon (RS) codes, where codewords are obtained as evaluations of suitably chosen polynomials. These codes were introduced by Tamo and Barg (IEEE Trans Inf Theory 60(8):4661–4676, 2014) where they assumed that the length n of the code is divisible by $$r+1$$, where r is the locality of the code. They also proposed a construction with this condition lifted to $$n \ne 1 \bmod (r+1)$$. In a recent paper, Kolosov et al. (Optimal LRC codes for all lenghts $$n \le q$$, arXiv:1802.00157, 2018) have given an explicit construction of optimal LRC codes with this lifted condition on n. In this paper we remove any such restriction on n completely, i.e., we propose constructions for q-ary RS-like LRC codes of any length $$n \le q$$. Further, we show that the codes constructed by the proposed construction are optimal LRC codes for their parameters.

For a Distributed Storage System (DSS), the Fractional Repetition (FR) code is a class in which replicas of encoded data packets are stored on distributed chunk servers, where the encoding is done using the Maximum Distance Separable (MDS) code. The FR codes allow for the exact uncoded repair with minimum repair bandwidth. In this paper, FR codes (called Flower codes) are constructed using finite binary sequences. It is shown that, for any FR code, there exists a Flower code and therefore Flower code is the general framework to construct FR code with uniform as well as non-uniform parameters. The condition for universally good Flower code is calculated on such sequences. For some sequences, the universally good Flower codes and Locally Repairable Flower codes are explored. In addition, conditions for equivalent Flower codes and dual Flower codes are also investigated in this paper. Some families of Flower codes with non-uniform parameters are obtained such that, from those families, Flowers code with uniform parameters are optimal FR codes in the literature. It is shown that any FR code is a Flower code and some known FR codes are obtained as the special cases of Flower codes using sequences.

In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block $$a_0a_1\ldots a_{n-1}$$ is extended into a block $$a_0a_1\ldots a_{n-1}d_0d_1\ldots d_{n-1}$$, where the redundant characters $$d_0, d_1, \ldots , d_{n-1}$$ are defined with $$d_i=a_i*a_{i+1}*a_{i+2}$$, where $$*$$ is a linear quasigroup operation and the operations in the indexes are modulo n. We give a proof that under some conditions the code is linear. Using this fact, we contribute to the determination of the error-detecting capability of the code. Namely, we determine the Hamming distance of the code and from there we obtain the number of errors that the code will detect for sure when linear quasigroups of order 4 from the best class of quasigroups of order 4 for which the constant term in the linear representation is zero matrix are used for coding. All results in the paper are derived for arbitrary length of the input blocks. With the obtained results we showed that when a small linear quasigroup of order 4 from the best class of quasigroups of order 4 is used for coding, the number of errors that the code surely detects is upper bounded with 4.

In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power $q$ with $q$ square, the total number of lengths for MDS self-dual codes over $\mathbb{F}_q$ presented in this paper is much more than those in all the previous results.

A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartlomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. The case for the regular pentagon is more challenging, and has resisted a completely analytic attack. In this paper, we present a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software. Our proof has a short proof certificate, is checkable without specialized software and is adaptable to other situations.

In 2014, Tamo and Barg have presented in a very remarkable paper a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality). The key ingredients for constructing such optimal linear LRC codes are the so-called r-good polynomials, where r is equal to the locality of the LRC code. In 2018, Liu et al. presented two general methods of designing r-good polynomials by using function composition, which led to three new constructions of r-good polynomials. Next, Micheli provided a Galois theoretical framework which allows to construct r-good polynomials. The well-known Dickson polynomials form an important class of polynomials which have been extensively investigated in recent years in different contexts. In this paper, we provide new methods of designing r-good polynomials based on Dickson polynomials. Such r-good polynomials provide new constructions of optimal LRC codes.

It is known that Julia sets, as a rule, have a fractal structure. In this paper, we give examples of smooth Julia sets, among them: a circle, a segment, an infinite interval, a straight line, and the complex plane. It is shown that the functions studied in the paper are chaotic on their Julia sets. The results obtained by analytical research are visualized using computer programs. The algorithms for constructing the Julia sets being considered are indicated.

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable $${{\mathcal {I}}}$$. Homeostasis plays an important role in the regulation of biological systems, cf. Ferrell (Cell Syst 2:62–67, 2016), Tang and McMillen (J Theor Biol 408:274–289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to $${{\mathcal {I}}}$$ is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387–407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input–output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1–24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input–output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input–output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760–773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305–364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264–275, 2014. https://doi.org/10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104–110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma et al. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities.

This paper proposes an algorithm to reason on constraints expressed in terms of polynomials with integer coefficients whose variables take values from finite subsets of the integers. The proposed algorithm assumes that an initial approximation of the domains of variables is available in terms of a bounding box, and it recursively subdivides the box into disjoint boxes until a termination condition is met. The algorithm includes three termination conditions that allow using it for three related reasoning tasks: constraint satisfaction, enumeration of solutions, and hyper-arc consistency enforcement. Considered termination conditions are based on suitable lower and upper bounds for polynomial functions over boxes that are determined using new results proved in the paper. The algorithm is particularly appropriate to reason on high-degree polynomial constraints because the proposed method to determine lower and upper bounds can outperform alternative methods when high-degree polynomials in a moderate number of variables are considered.

According to the P conjecture by Gh. Păun, polarizationless P systems with active membranes cannot solve $${\mathbf {NP}}$$NP-complete problems in polynomial time. The conjecture is proved only in special cases yet. In this paper we consider the case where only elementary membrane division and dissolution rules are used and the initial membrane structure consists of one elementary membrane besides the skin membrane. We give a new approach based on the concept of object division polynomials introduced in this paper to simulate certain computations of these P systems. Moreover, we show how to compute efficiently the result of these computations using these polynomials.

Determine the number of the rational zeros of any given linearized polynomial is one of the vital problems in finite field theory, with applications in modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact number when applied to a specific linearized polynomial. The first contribution of this paper is a better general method to get a more precise upper bound on the number of rational zeros of any given linearized polynomial over arbitrary finite field. We anticipate this method would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second-order nonlinearities of general cubic Boolean functions, which has been an active research problem during the past decade. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean function one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean functions $g_{\mu }=Tr(\mu x^{2^{2r}+2^{r}+1})$ defined over the finite field ${\mathbb F}_{2^{n}}$ .

In this paper we propose a generalized Roman domination problem called connected strong k-Roman dominating set problem. It is NP-hard even in a unit ball graph. Unit ball graphs are the intersection graphs of equal sized balls in the three-dimensional space, they are widely used as a mathematical model for wireless sensor networks and some problems in computational geometry. This paper presents the first constant approximation algorithm with a guaranteed performance ratio at most $$6(k+2)$$ in unit ball graphs, where k is a positive integer.

Linear codes with complementary duals (shortly named LCD codes) are linear codes whose intersection with their duals are trivial. In this paper, we give a method of constructing these type of linear codes from equitable partitions of association schemes. The LCD codes constructed in this paper are of length 2n and dimension n and have the property of being formally self-dual. To illustrate the method we construct LCD codes from some distance-regular graphs.

The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In 1991, C. Jockusch and R. Soare constructed a low linear order without computable presentations. Earlier, in 1989, R. Downey and M. Moses showed that each low discrete linear order has a computable copy. It is natural to ask for which order types of low presentations the existence of a computable presentation is sufficient. This question (namely, research program) was stated by R. Downey in 1998: Describe the order property P such that, for any low linear order L, P(L) implies the existence of a computable presentation of L. In this paper, we give a detailed review of the main results in this direction. These results are mostly obtained by the author or in co-authorship.